Model Theory of Generic Vector Space Endomorphisms

TL;DR

This paper develops a model-theoretic framework for endomorphisms of vector spaces, characterizing existentially closed models and conditions for the existence of model companions in this setting.

Contribution

It introduces a systematic approach to analyze endomorphisms with additional structure on vector spaces, including criteria for model companions and first-order expressibility.

Findings

Characterization of existentially closed models of endomorphisms

Parametrization of consistent extensions involving polynomial conditions

Sufficient criteria for the existence of model companions

Abstract

This paper deals with the model companion of an endomorphism acting on a vector space, possibly with extra structure. Given a theory $T$ that $\varnothing$-defines an infinite $K$-vector space $\mathbb{V}$ in every model, we set $T_\theta := T \cup \{\text{``$\theta$ defines a $K$-endomorphism of $\mathbb{V}$"}\}$. We then consider extensions of the form $$ T_\theta \cup \left\{\sum\nolimits_{k}\bigcap\nolimits_{l}\operatorname{Ker}(\rho_{j, k, l}[\theta]) = \sum\nolimits_{k}\bigcap\nolimits_{l} \operatorname{Ker}(\eta_{j, k, l}[\theta]) : j \in \mathcal{J}\right\}, $$ where all sums and intersections are finite, and all the $\rho[\theta]$'s and $\eta[\theta]$'s are polynomials over $K$ with $\theta$ plugged in. Notice that properties such as $\theta^2 - 2\operatorname{Id} = 0$ or $\operatorname{Ker}(\theta^n) = \operatorname{Ker}(\theta^{n+1})$ can be expressed in such a manner. We then parametrize the consistent extensions of this form by a family $\{T^C_\theta : C \in \mathcal{C}\}$ and characterize the existentially closed models of each $T^C_\theta$. We also present a sufficient criterion, which only depends on $T$, for when these characterizations are first-order expressible, i.e., for when a model companion of each $T^C_\theta$ exists.